x^2+(x^2)/(x+1)^2=3
Find the values of x
x^2 + x^2/(x+1)^2 = 3
times (x+1)^2
x^2(x+1)^2 + x^2 = 3(x+1)^2
x^2(x^2 + 2x + 1) + x^2 = 3(x^2 + 2x + 1)
x^4 + 2x^3 + x^2 + x^2 = 3x^2 + 6x + 3
x^4 + 2x^3 - x^2 - 6x - 3 = 0
nasty equation, so I tried Wolfram,
http://www.wolframalpha.com/input/?i=x%5E4+%2B+2x%5E3+-+x%5E2+-+6x+-+3+%3D+0
and to my surprise, the answer is
(1 ± √5)/2
The positive result just happens to be the "Golden Ratio".
Google "Golden Ratio" and open up a life-time lasting topic of Mathematics
e.g. notice that when you use your calculator to evaluate (1+√5)/2 to get 1.618033989..
and then square it, the decimal part will not change. No other number behaves like that.
Just one of hundreds of properties of that number.
But I digress .